It is known (see for example Friz-Victoir), for a Gaussian process $X$ that is $\alpha$-Hoelder for $\alpha>1/4$, that the canonical rough path $(\int dX,\int\int dX\otimes dX,\int\int\int dX\otimes dX\otimes dX)$ satisfies a Freidlin-Wentzell LDP.
I'm curious, if we replace the integrand $1$ with deterministic $L^2$ function $f$ (maybe even assume that it is smooth) so we have e.g. $\int\int f ~dX\otimes dX$ do we also get a Freidlin-Wentzell type LDP? I am fine for each individual object and not the triple, too.
